Flight engineer used to be the path to the right seat. Coming from a position of no experience, however, into the right seat, is tough, especially given that you've got zero experience, to start.
The siren song of a job that avoids instructing is a very common theme. I'm different. I don't want to instruct. I've put in my dues. I did a whopping xxx hours in general aviation, and many other tunes of entitlement.
What you need isn't time in a clapped out airplane adjusting the landing pressurization and reconnecting the split system breaker for the umpteenth time. It won't teach you about flying. It won't move your career along. You're looking at pading your logbook with time to circumvent a job where you'd actually be learning about flying.
I've known a number of FE's who attempted to upgrade, but who were so atrophied in the third seat that they couldn't remember how to fly, and failed their training. Most of them were great flight engineers, too, but weak pilots.
I'm a flight engineer, among other stuff; I'm not unsympathetic, and I like a cockpit that has a flight engineer. I think they're a valued crewmember and a great resource, albeit rapidly becoming extinct.
If you really want to learn airplane systems, become a mechanic.
If you really want to fly, be a pilot.